[FOM] Utilitarian critiques/analyses/surveys of mathematical methodology

[Richard Haney]

I can think of several obvious advantages and disadvantages to the use
of deductive logic and the axiomatic method in mathematics.  Among the
advantages are objectivity, elimination of confusion, (at least for
many thinkers) elimination of contradictions, and the intuitiveness of
many deductions as being in some sense "correct" and productive.  Among
the disadvantages are the well-known "pathologies" of mathematics, the
existence of non-intuitive or even counter-intuitive results, the
existence of exceedingly intricate, exceedingly long proofs spanning
several books, and often the seeming irrelevance of mathematics to
reality.

So, in this vein, does anyone know of any "completely open-minded",
"no-holds-barred", strictly utilitarian critiques/analyses/surveys of
mathematical methodology of a foundational nature?  By
"no-holds-barred" I mean that no mathematical ideology is held sacred,
including the ideology that there are some rules of inference that are
infallible for producing "new knowledge" (or "new results", for those
who might want to quibble as to what knowledge is).

As usual, specific resources (including sections and page numbers, if
available) would be appreciated.  I am particularly interested in
critiques/analyses/surveys that are especially mindful of the
usefulness of mathematics for real-world applications.

Richard Haney


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